Local isometric immersions of pseudospherical surfaces described by a class of third order partial differential equations

2025/05/26 by Guo, Mingyue, Shi, Zhenhua
#FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2505.19728

Abstract

In this paper, we study the problem of local isometric immersion of pseudospherical surfaces determined by the solutions of a class of third order nonlinear partial differential equations with the type u_t - u_xxt = λu² u_xxx + G(u, u_x, u_xx),(λ∈ℝ). We prove that there is only two subclasses of equations admitting a local isometric immersion into the three dimensional Euclidean space 𝔼³ for which the coefficients of the second fundamental form depend on a jet of finite order of u, and furthermore, these coefficients are universal, namely, they are functions of x and t, independent of u. Finally, we show that the generalized Camassa-Holm equation describing pseudospherical surfaces has a universal second fundamental form.

Citations

• On a class of third order differential equations describing pseudospherical or spherical surfaces
• On a class of systems of hyperbolic equations describing pseudo-spherical or spherical surfaces
• Local isometric immersions of pseudo-spherical surfaces and k-th order evolution equations
• Third order differential equations and local isometric immersions of\n pseudospherical surfaces
• Local Isometric immersions of pseudo-spherical surfaces and evolution equations
• Second-order equations and local isometric immersions of\n pseudo-spherical surfaces

Related

• Local isometric immersions of pseudospherical surfaces described by a class of third order differential equations
• Third order differential equations and local isometric immersions of\n pseudospherical surfaces
• A note on isometric immersions and differential equations which describe\n pseudospherical surfaces
• Second-order equations and local isometric immersions of\n pseudo-spherical surfaces
• Pseudo-potentials and local isometric immersion for a generalized Camassa-Holm equation describing pseudospherical surfaces